Concept-based methods have emerged as a promising direction to develop interpretable neural networks in standard supervised settings. However, most works that study them in incremental settings assume either a static concept set across all experiences or assume that each experience relies on a distinct set of concepts. In this work, we study concept-based models in a more realistic, dynamic setting where new classes may rely on older concepts in addition to introducing new concepts themselves. We show that concepts and classes form a complex web of relationships, which is susceptible to degradation and needs to be preserved and augmented across experiences. We introduce new metrics to show that existing concept-based models cannot preserve these relationships even when trained using methods to prevent catastrophic forgetting, since they cannot handle forgetting at concept, class, and concept-class relationship levels simultaneously. To address these issues, we propose a novel method - MuCIL - that uses multimodal concepts to perform classification without increasing the number of trainable parameters across experiences. The multimodal concepts are aligned to concepts provided in natural language, making them interpretable by design. Through extensive experimentation, we show that our approach obtains state-of-the-art classification performance compared to other concept-based models, achieving over 2x the classification performance in some cases. We also study the ability of our model to perform interventions on concepts, and show that it can localize visual concepts in input images, providing post-hoc interpretations.